It can also be a good career move. A (likely apocryphal) story goes: when Peter Lax was awarded the National Medal of Science, the other recipients (presumably non-mathematicians) asked him what he did to deserve the Medal. Lax responded: "I integrated by parts."
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Willie WongApr 24 '11 at 23:42

Two more stories: 1. Supposedly when Laurent Schwartz received the Fields Medal (for his work on distributions, of course), someone present remarked, "So now they're giving the Fields Medal for integration by parts." 2. I believe I remember reading -- but have no idea where -- that someone once said that a really good analyst can do marvelous things using only the Cauchy-Schwarz inequality and integration by parts. I do think there's some truth to that.
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Carl OffnerOct 11 '11 at 2:15

Highbrow: Derivation of the Euler-Lagrange equations describing how a physical system evolves through time from Hamilton's Least Action Principle.

Here's a very brief summary. Consider a very simple physical system consisting of a point mass moving under the force of gravity, and suppose you know the position $q$ of the point at two times $t_0$ and $t_f$. Possible trajectories of the particle as it moved from its starting to ending point correspond to curves $q(t)$ in $\mathbb{R}^3$.

One of these curves describes the physically-correct motion, wherein the particle moves in a parabolic arc from one point to the other. Many curves completely defy the laws of physics, e.g. the point zigs and zags like a UFO as it moves from one point to the other.

Hamilton's Principle gives a criteria for determining which curve is the physically correct trajectory; it is the curve $q(t)$ satisifying the variational principle

$$\min_q \int_{t_0}^{t_f} L(q, \dot{q}) dt$$
subject to the constraints $q(t_0) = q_0, q(t_f) = q_f$.
Where $L$ is a scalar-valued function known as the Lagrangian that measures the difference between the kinetic and potential energy of the system at a given moment of time. (Pedantry alert: despite being historically called the "least" action principle, really instead of minimizing we should be extremizing; ie all critical points of the above functional are physical trajectories, even those that are maxima or saddle points.)

It turns out that a curve $q$ satisfies the variational principle if and only if it is a solution to the ODE
$$ \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} + \frac{\partial L}{\partial q} = 0,$$
roughly equivalent to the usual Newton's Second Law $ma-F=0$, and the key step in the proof of this equivalence is integration by parts. What is remarkable here is that we started with a boundary-value problem -- given two positions, how did we get from one to the other? -- and ended with an ODE, an initial-value problem -- given an initial position and velocity, how does the point move as we advance through time?

Highbrow: Integration by parts can be used to compute (or verify) formal adjoints of differential operators.
For instance, one can verify, and this was indeed the proof I saw, that the formal adjoint of the Dolbeault operator $\bar{\partial}$ on complex manifolds is $$\bar{\partial}^* = -* \bar{\partial} \,\,\, *, $$
where $*$ is the Hodge star operator, using integration by parts.

This is also called Wallis formula/product I believe.
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AryabhataFeb 5 '11 at 18:04

@Aryabhata Yes. This would've been more interesting is he showed how to get it. It's not too hard.
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Pedro Tamaroff♦Feb 23 '12 at 16:58

@PeterT.off: Are you talking about the infinite version? He did show the finite version.
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AryabhataFeb 23 '12 at 17:00

@Aryabhata I've never see the Wallis finite product. I always seen Walli's infinite product. I guess it'd be better to at least hint what $\dfrac{I_{2k+1}}{I_{2k}}$ is, and that it tends to 1.
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Pedro Tamaroff♦Feb 23 '12 at 17:05

Hm, this example does not depend on integration by parts so much as it depends on not keeping track of the limits of integration.
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Greg GravitonFeb 4 '11 at 16:05

It's true that the crux of the problem is not so much in the integration by parts, but if you integrate in a different way (what way, by the way?) you won't have that problem.
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RaskolnikovFeb 4 '11 at 19:40

13

@Greg: Actually, it's not the limits of integration that matter here, but the constant of integration. $\int \frac{1}{x}\,dx$ is the entire family of antiderivatives, which is exactly the same as the family you get if you add $1$ to every member of the family.
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Arturo MagidinFeb 4 '11 at 22:08

My favorite example is getting an asymptotic expansion: for example, suppose we want to compute $\int_x^\infty e^{-t^2}\cos(\beta t)dt$ for large values of $x$. Integrating by parts multiple times we end up with
$$ \int_x^\infty e^{-t^2}\cos(\beta t)dt
\sim e^{-x^2}\sum_{k=1}^\infty(-1)^n\frac{H_{k-1}(x)}{\beta^k}
\begin{cases}
\cos(\beta x) & k=2n \\
\sin(\beta x) & k=2n+1
\end{cases}$$
where the Hermite polynomials are given by
$H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$.

This expansion follows mechanically applying IBP multiple times and gives a nice asymptotic expansion (which is divergent as a power series).

Integrating by parts is the how one discovers the adjoint of a differential operator, and thus becomes the foundation for the marvelous spectral theory of differential operators. This has always seemed to me to be both elementary and profound at the same time.